SECTION A.5 Rational Expressions A35 5. True or False In using synthetic division, the divisor is always a polynomial of degree 1, whose leading coefficient is 1. 6. True or False )2 5 3 2 1 10 14 32 5 7 16 31 − − − − − means + + + + = − + + − + x x x x x x x 5 3 2 1 2 5 7 16 31 2 . 3 2 2 Skill Building In Problems 7–18, use synthetic division to find the quotient and remainder when: 7. − + + x x x 7 5 10 3 2 is divided by −x 2 8. + − + x x x 2 3 1 3 2 is divided by +x 1 9. + − + x x x 3 2 3 3 2 is divided by −x 3 10. − + − + x x x 4 2 1 3 2 is divided by +x 2 11. − + x x x 4 5 3 is divided by +x 3 12. + + x x 2 4 2 is divided by −x 2 13. − + + x x x 4 3 5 6 4 2 is divided by −x 1 14. + − x x5 10 5 3 is divided by +x 1 15. + x x 0.1 0.2 3 is divided by +x 1.1 16. − x 0.1 0.2 2 is divided by +x 2.1 17. − x 32 5 is divided by −x 2 18. + x 1 5 is divided by +x 1 In Problems 19–28, use synthetic division to determine whether −x c is a factor of the given polynomial. 19. − − + − x x x x 4 3 8 4; 2 3 2 20. − + + + x x x 4 5 8; 3 3 2 21. − − + − x x x x 2 6 7 21; 3 4 3 22. − − − x x x 4 15 4; 2 4 2 23. + + + x x x 5 43 24; 2 6 3 24. − + − + x x x x 2 18 9; 3 6 4 2 25. − − + + x x x x 16 19; 4 5 3 2 26. − + − + x x x x 16 16; 4 6 4 2 27. − + − − x x x x 3 6 2; 1 3 4 3 28. + − + + x x x x 3 3 1; 1 3 4 3 Applications and Extensions 29. Find the sum of a , b , c , and d if − + + + = + + + + x x x x ax bx c d x 2 3 5 2 2 3 2 2 Explaining Concepts 30. When dividing a polynomial by −x c, do you prefer to use long division or synthetic division? Does the value of c make a difference to you in choosing? Give reasons. 1 Reduce a Rational Expression to Lowest Terms If we form the quotient of two polynomials, the result is called a rational expression . Some examples of rational expressions are (a) + x x 1 3 (b) + − + x x x 3 2 5 2 2 (c) − x x 1 2 (d) ( ) − xy x y 2 2 Expressions (a), (b), and (c) are rational expressions in one variable, x , whereas (d) is a rational expression in two variables, x and y . A.5 Rational Expressions OBJECTIVES 1 Reduce a Rational Expression to Lowest Terms (p. A35) 2 Multiply and Divide Rational Expressions (p. A36) 2 Add and Subtract Rational Expressions (p. A37) 4 Use the Least Common Multiple Method (p. A39) 5 Simplify Complex Rational Expressions (p. A40)
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